What kind of network your time series becomes when you connect every pair of points that can "see" each other over the intervening values.
Treats the signal as a landscape and draws an edge between two time points whenever no intermediate value blocks the line of sight between them. Tall peaks see far; valleys are hidden. The resulting graph's degree distribution, clustering, and assortativity encode the signal's dynamical class. Periodic signals produce regular graphs. Chaotic signals produce scale-free networks. Random signals produce specific power-law exponents predicted by theory.
Do high-degree nodes connect to other high-degree nodes (positive), or to low-degree ones (negative)? Rössler Hyperchaos (0.76) is strongly assortative: its tall peaks cluster together. LIGO Livingston (0.54) also shows this pattern. Lotka-Volterra (-0.57) is strongly disassortative: its predator-prey oscillations create peaks that connect to valleys, not to each other. Logistic Period-2 (-0.50) alternates high-low, producing the same disassortative pattern.
How interconnected are each node's neighbors? Rainfall (0.85) has the highest clustering -- its rain events create local clusters of mutually visible points. Neural Net Pruned (0.83) and Sensor Event Stream (0.81) show similar cliquish structure. Devil's Staircase (0.0) has zero clustering because its flat plateaus followed by jumps create a graph where neighbors never see each other.
Power-law exponent of the degree distribution. DNA SARS-CoV-2 (2.94) has the steepest decay -- most nodes have low degree, with rare high-degree hubs. Circle Map QP (2.78) and Rule 30 (2.77) follow. For iid random data, the NVG degree distribution is exponential (Lacasa et al. 2008), so the power-law exponent from a forced fit is not theoretically meaningful in that regime — but deviations across signals still reveal structural differences.
The most-connected node in the graph. Lotka-Volterra (690) has an extreme hub -- its largest predator-prey peak can see across 690 other time points. Rainfall (587) and Van der Pol (540) also produce high-visibility peaks. Devil's Staircase has max_degree = 2, meaning no point can see beyond its immediate neighbors.
How well does a power law fit the degree distribution? Earthquake Depths (0.97) and Poker Hands (0.96) have nearly perfect power-law degree distributions. Devil's Staircase (0.0) has no power-law structure (its flat plateaus create a degenerate degree distribution). Sine Wave (0.26) scores low because its regular oscillations create a peaked, non-power-law distribution. High R² confirms that the visibility graph has genuine scale-free topology.
Fraction of all possible edges that exist. Sine Wave (0.156) has the densest visibility graph — its smooth oscillations allow long-range visibility. Lotka-Volterra (0.145) and Duffing (0.137) are close behind. Devil's Staircase (0.002) and Forest Fire (0.002) are the sparsest — their flat regions block almost all visibility lines. This measures the global "openness" of the signal landscape.
Jensen-Shannon divergence between the edge sets of the natural visibility graph (NVG) and the horizontal visibility graph (HVG). Lotka-Volterra (0.98), Sine Wave (0.98), and Duffing (0.98) score highest — their smooth peaks create many NVG edges that the stricter HVG misses. Fibonacci Word and Devil's Staircase score 0.0 (NVG and HVG produce identical graphs for binary or staircase-like signals). This measures how much "diagonal" visibility the signal has beyond horizontal visibility.
Divergence between NVG and HVG in terms of how far each node can "see" (average reach per node). Exponential Chirp (1.0), PID Controller (1.0), and Lotka-Volterra (1.0) score highest. Fibonacci Word and Devil's Staircase score 0.0. This captures differences in long-range visibility structure between the two graph types.
| Source | Domain | Value |
|---|---|---|
| Rössler Hyperchaos | chaos | 0.7641 |
| LIGO Livingston | astro | 0.5444 |
| Clipped Sine | waveform | 0.4930 |
| ··· | ||
| Lotka-Volterra | bio | -0.5740 |
| Logistic r=3.2 (Period-2) | chaos | -0.4966 |
| Hodgkin-Huxley | bio | -0.4472 |
| Source | Domain | Value |
|---|---|---|
| Rainfall (ORD Hourly) | climate | 0.8602 |
| Neural Net (Pruned 90%) | binary | 0.8300 |
| Sensor Event Stream | exotic | 0.8124 |
| ··· | ||
| Devil's Staircase | exotic | 0.0000 |
| Forest Fire | exotic | 0.0906 |
| Triangle Wave | waveform | 0.1910 |
| Source | Domain | Value |
|---|---|---|
| Rule 30 | exotic | 2.7720 |
| DNA SARS-CoV-2 | bio | 2.7679 |
| Collatz Parity | number_theory | 2.7659 |
| ··· | ||
| Devil's Staircase | exotic | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.4214 |
| Sine Wave | waveform | 0.5497 |
| Source | Domain | Value |
|---|---|---|
| Earthquake Depths | geophysics | 0.9743 |
| Poker Hands | exotic | 0.9604 |
| Collatz Stopping Times | number_theory | 0.9587 |
| ··· | ||
| Devil's Staircase | exotic | 0.0000 |
| Damped Pendulum | motion | 0.2383 |
| Clipped Sine | waveform | 0.2561 |
| Source | Domain | Value |
|---|---|---|
| Sine Wave | waveform | 0.1564 |
| Lotka-Volterra | bio | 0.1451 |
| Duffing Oscillator | chaos | 0.1366 |
| ··· | ||
| Devil's Staircase | exotic | 0.0020 |
| Forest Fire | exotic | 0.0024 |
| Fibonacci Word | exotic | 0.0027 |
| Source | Domain | Value |
|---|---|---|
| Lotka-Volterra | bio | 690.4000 |
| Rainfall (ORD Hourly) | climate | 614.4400 |
| PID Controller | exotic | 547.4000 |
| ··· | ||
| Devil's Staircase | exotic | 2.0000 |
| Fibonacci Word | exotic | 4.0000 |
| Logistic r=3.2 (Period-2) | chaos | 4.0000 |
| Source | Domain | Value |
|---|---|---|
| Lotka-Volterra | bio | 0.9844 |
| Sine Wave | waveform | 0.9832 |
| Duffing Oscillator | chaos | 0.9793 |
| ··· | ||
| Fibonacci Word | exotic | 0.0000 |
| Logistic Edge-of-Chaos | chaos | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Exponential Chirp | exotic | 0.9990 |
| PID Controller | exotic | 0.9986 |
| Lotka-Volterra | bio | 0.9985 |
| ··· | ||
| Devil's Staircase | exotic | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
Visibility Graph is the atlas's best tool for separating oscillatory dynamics by their waveform shape. The assortativity metric spans from -0.57 to +0.76 across the atlas, a range no other metric covers, and it cleanly separates predator-prey / period-doubling dynamics (negative) from hyperchaotic / bursty dynamics (positive). The clustering coefficient uniquely identifies bursty event streams and rainfall as topologically cliquish -- a property invisible to spectral or distributional geometries.